Link to actual problem [6576] \[ \boxed {\cos \left (x \right ) y^{\prime \prime }+y^{\prime }+5 y=0} \] With the expansion point for the power series method at \(x = 0\).
type detected by program
{"second order series method. Ordinary point", "second order series method. Taylor series method"}
type detected by Maple
[[_2nd_order, _with_linear_symmetries]]
Maple symgen result This shows Maple’s found \(\xi ,\eta \) and the corresponding canonical coordinates \(R,S\)\begin{align*} \\ \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \operatorname {HeunG}\left (\frac {1}{2}, 5 i, 0, 0, 1, -1, \frac {\left (-\frac {1}{2}+\frac {i}{2}\right ) \left (\tan \left (\frac {x}{2}\right )+1\right )}{-\tan \left (\frac {x}{2}\right )+i}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {y}{\operatorname {HeunG}\left (\frac {1}{2}, 5 i, 0, 0, 1, -1, \frac {\left (-1+i\right ) \left (\tan \left (\frac {x}{2}\right )+1\right )}{-2 \tan \left (\frac {x}{2}\right )+2 i}\right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \operatorname {HeunG}\left (\frac {1}{2}, 5 i, 0, 0, 1, -1, \frac {\left (-\frac {1}{2}+\frac {i}{2}\right ) \left (\tan \left (\frac {x}{2}\right )+1\right )}{-\tan \left (\frac {x}{2}\right )+i}\right ) \left (\int \frac {\left (\tan \left (\frac {x}{2}\right )-1\right ) \left (\cos \left (x \right )+1\right ) \left (1+\tan \left (\frac {x}{2}\right )^{2}\right )}{4 \left (\tan \left (\frac {x}{2}\right )+1\right ) \operatorname {HeunG}\left (\frac {1}{2}, 5 i, 0, 0, 1, -1, \frac {\left (-\frac {1}{2}+\frac {i}{2}\right ) \left (\tan \left (\frac {x}{2}\right )+1\right )}{-\tan \left (\frac {x}{2}\right )+i}\right )^{2}}d x \right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {y}{\operatorname {HeunG}\left (\frac {1}{2}, 5 i, 0, 0, 1, -1, \frac {\left (-1+i\right ) \left (\tan \left (\frac {x}{2}\right )+1\right )}{-2 \tan \left (\frac {x}{2}\right )+2 i}\right ) \left (\int \frac {\left (\tan \left (\frac {x}{2}\right )-1\right ) \left (\cos \left (x \right )+1\right ) \left (1+\tan \left (\frac {x}{2}\right )^{2}\right )}{4 \left (\tan \left (\frac {x}{2}\right )+1\right ) \operatorname {HeunG}\left (\frac {1}{2}, 5 i, 0, 0, 1, -1, \frac {\left (-1+i\right ) \left (\tan \left (\frac {x}{2}\right )+1\right )}{-2 \tan \left (\frac {x}{2}\right )+2 i}\right )^{2}}d x \right )}\right ] \\ \end{align*}